Composites: Part B 40 (2009) 337–348

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Composites: Part B

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Size effect on strength of laminate-foam sandwich plates: Finite elementanalysis with interface fracture

Ferhun C. Caner a,*, Zdenek P. Bazant b

a Institute of Energy Technologies (INTE), School of Industrial Engineering (ETSEIB-UPC), Diagonal 647, E-08028 Barcelona, Spainb Civil Engineering and Materials Science, Northwestern University, 2145 Sheridan Road, CEE/A135, Evanston, IL 60208, USA

a r t i c l e i n f o

Article history:Received 31 March 2009Accepted 31 March 2009Available online 5 April 2009

Keywords:Size effectA. Sandwich plateB. Interface fractureA. Polymer composite

1359-8368/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.compositesb.2009.03.005

* Corresponding author.E-mail addresses: ferhun.caner@upc.edu (F.C. Can

edu (Z.P. Bazant).

a b s t r a c t

Recent three-point bend tests of size effect on the strength of geometrically scaled sandwich beams ofthree types – with no notches, and with notches at the upper or lower skin–foam interface, which werepreviously evaluated using simplified sandwich beam theory and equivalent linear elastic fracturemechanics, are now reanalyzed more accurately by finite elements. Zero-thickness interface elementswith a softening cohesive law are used to model fractures at the skin–foam interface, in the fiber com-posite skins, and in the foam. The fracture energy and fracture process zone length of a shear crack infoam near the interface are deduced by fitting an analytical expression for size effect to the test data.Numerical simulations reveal that small-size specimens with notches just under the top skin developplastic zones in the foam core near the edges of the loading platen, and that small-size specimens withnotches just above the bottom skin develop distributed quasibrittle fracture in the foam core under ten-sion. Both phenomena, though, are found to reduce the maximum load by less than 6%. Further it isshown that, in notch-less beams, the interface shear fracture is coupled with compression crushing ofthe fiber–polymer composite skin. For small specimens this mechanism is important because, when itis blocked in simulations, the maximum load increases. The size effect law for notch-less beams is cali-brated such that beams of all sizes fail solely by interface shear fracture.

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1. Introduction including [14]: (1) the debonding shear fracture of skins (or face

According to the current design practice, the strength of sand-wich structures is determined by means of plastic limit analysisor elastic analysis with a strength limit [1,38,20,23,25,22]. Thispractice ignores fracture mechanics and misses the deterministicsize effect. However, recent experiments at NorthwesternUniversity demonstrate that sandwich components, such as thefiber–polymer composites [29,30,35,26] and polymeric foams [8],usually fail in a brittle manner and exhibit strong deterministicsize effect. Obviously, the same must be expected for sandwichplates. This is particularly important for large sandwich structuressuch as the hulls, decks, bulkheads, masts and antenna covers oflarge ships, as well as for load-bearing fuselage panels, verticalstabilizers, rudders and wing boxes of aircraft.

The type and intensity of size effect depend on the failure mech-anism, i.e., different failure mechanisms require different size ef-fect laws. Thus, in testing for size effect, it is desirable that thespecimens of all sizes fail exhibit the same failure mechanism.Laminate-foam sandwich plates can fail in a variety of modes,

ll rights reserved.

er), z-bazant@northwestern.

sheets), (2) fracture in the foam core, (3) compression fracture ofskins [9], (4) delamination followed by skin buckling, (5) skinbuckling or wrinkling [19] followed by delamination [3], and (6)foam core indentation. These modes often interact. The first threemodes must, in principle, lead to deterministic size effects, and somust, at least partly, the fourth and fifth. Recent experiments re-ported in [6] were designed so as to obtain failure in one or bothof the first two modes, although compression fracture of skinsaccompanied some of the failures.

Some previous studies treated skin debonding in terms of linearelastic fracture mechanics (LEFM) [36,37]. However, LEFM impliesthat the fracture process zone (FPZ) size shrinks to a point and thebehavior is brittle. Recent size effect experiments on fiber–polymerlaminates and on PVC foam, conducted at Northwestern University[7,8] reveal that the size effect is weaker than the power-law sizeeffect of LEFM, implying quasibrittle behavior in which the FPZ sizecannot be neglected. A numerical demonstration of size effect in sand-wich plates with a limited size range, failing by core indentation, wasreported in [10]. In that study, a simple constitutive law with soften-ing (derived from pore collapse) was considered for the core, while theskin was assumed to have a plastic strength limit. Recently, a morecomprehensive investigation of the size effect in sandwich plates,both experimental and analytical, was undertaken [6].

338 F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348

This study is focussed on sandwich plates made up of compositelaminate skins and PVC foam core, which are used in the construc-tion of various structural parts of ships and aircraft. The objectivesare three: (i) to determine the shear fracture energy of the poly-meric foam, (ii) to find out whether interface fracture was the onlyfailure mechanism in Series I and II specimens that were used in[6] to validate the size effect laws; and (iii) to identify, for the caseswith multiple failure mechanisms, the size effect law that corre-sponds to interface fracture alone. To this end, accurate finite ele-ment analyses of the test specimens are performed to fit theexperimental data. Zero-thickness interface elements are insertedalong the experimentally observed fracture surfaces. A cohesivefracturing law for opening and shear is used to describe the behav-ior of the foam core, and localized compression-shear crushing ofthe fiber–polymer composite skins is also taken into account.

First, a brief description of the size effect tests of the sandwichbeams will be given. Following the description of the finite elementanalyses performed, the size effect laws for notched and notch-less

Fig. 1. Dimensions of test beams with (a) notches at top inte

sandwich plates will be formulated. Finally the results will be dis-cussed and conclusions will be drawn. The details of the experi-ments analyzed [6] are summarized in the Appendix.

2. Size effect tests of sandwich beams

As reported in [6], three series of beams geometrically scaled intwo dimensions were fabricated and tested in three-point bending:Series I, with beams notched symmetrically from both ends in thefoam next to the interface of bottom skin (or facing, face sheet);Series II. ditto but top skin; and Series III. ditto but un-notched(Fig. 1(a)–(c)). The failure mechanism was different in each series.The scaling ratios were 1:4:16 for series I, and 1:3:9 for series IIand III. The cores of sandwich beams were made of closed cell poly-vinylchloride (PVC) foam, and the skins were made of woven glass-epoxy composite (FS-12A) in series I and of unidirectional carbon-epoxy composite (IM6-G/3501-6) in series II and III. For test de-tails; see the Appendix and [6].

rface, (b) notches at bottom interface and (c) no notches.

F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348 339

The purpose of making the notches was to clarify the effect oflarge pre-existing cracks or damage zones, and to force the fractureto develop at a certain pre-determined location, homologous (i.e.,geometrically similar) for all the sizes. Forcing the cracks to startat a predetermined location eliminates the possibility of Weibull-type statistical contribution to the size effect [2,4], because theinitiating fracture cannot sample locations of different randommaterial strength. In that case, the only effect of material random-ness is a size-independent scatter of structural strength.

3. Finite element analysis of sandwich beams

3.1. The zero-thickness interface element

Cohesive shear fracture at the skin–foam interface dominatedthe failure in Series I and II, and was the only important failuremechanism in Series III. For finite element analysis of such failures,the so-called ‘zero-thickness’ interface element is a natural candi-date. This element was originally introduced [21] to analyzejointed rocks. It was also employed to represent cracks in concrete[13], delamination of polymer composites [17], fracture in particle-reinforced composites [32] and biological bones [12]. Similar for-mulations for the so-called ‘spring elements’ were also developed[33,34]. In [16], a cohesive finite element, more efficient than thetraditional continuum cohesive elements, was developed. For amore detailed literature review on interface elements, see [28].

This study uses an interface element that was coded (at UPC) inan implicit finite element program. The element is made up of twofour-node quadrilaterals, which connect the faces of two adjacenthexahedral continuum elements. The surfaces of the two quadrilat-erals are initially coincident, which means that initially the ele-ment has zero thickness. Later, these surfaces are allowed toseparate, to simulate crack opening, and their relative normaland shear displacements produce normal and shear stresses, cap-tured at Gauss points of finite elements according to the prescribedconstitutive law. In three dimensions, the interface element has 24global degrees of freedom given by the column vector u ¼ ðuIiÞwhere the index I varies from 1 to 8 and the index i varies from1 to 3. The hats label the quantities at the nodes, and the corre-sponding quantities without hat are those defined over the ele-ment; the nodal quantities are defined with respect to globalcoordinates unless otherwise is noted. The nodal relative displace-ment vector Du is computed as the difference in displacements be-tween the corresponding nodes [32]:

Du ¼ ðI12�12 � I12�12Þu ð1Þ

Here I is the identity matrix; Du ¼ ðDuIiÞ where I ¼ 1;2; . . . 4 andfor each I, i ¼ 1;2;3. The relative displacement field over the ele-ment can be interpolated as

Duðn;gÞ ¼ ð/1I3�3;/2I3�3;/3I3�3;/4I3�3ÞDu ð2Þ

where /I ¼ /Iðn;gÞ in which I ¼ 1;2; . . . 4 are the usual shape func-tions; and n and g are the parent coordinates.

When large displacements are considered, the local coordinatesxref of the interface element may conveniently be defined as thecoordinates of the midsurface between the two faces of the ele-ment [32]:

xref ¼12ðI12�12; I12�12Þðxþ uÞ ð3Þ

where x ¼ ðxIiÞ are the nodal coordinates (I ¼ 1;2; . . . ;8; and foreach I; i ¼ 1;2;3Þ; xref ¼ ðxref

Ii Þ where I ¼ 1;2; . . . 4; and for eachI; i ¼ 1;2;3. The coordinates of the midsurface could be interpolatedusing

x ðn;gÞ ¼ ð/ I ;/ I ;/ I ;/ I Þx ð4Þ

ref 1 3�3 2 3�3 3 3�3 4 3�3 ref

When the deformations are small, the reference surface definedabove becomes indistinguishable from the original surface. Next, alocal coordinate system can be defined over this midsurface usingthe vectors:

tn ¼@xref

@n1

j@xref =@nj

tg ¼@xref

@g1

j@xref =@gjð5Þ

n ¼ tn � tg

The rotation matrix that transforms vectors from the global coordi-nate system to the local one is given by

R ¼ n! tn ! tg !

264

375

T

ð6Þ

where ! marks row vectors.From the internal virtual work of an element,

dWint ¼ duT f eint ¼

Z@Xe

dDuT tdS ¼Z@Xe

ðUduÞT tdS ð7Þ

one obtains the nodal forces of the element:

f eint ¼

Z@Xe

UT tdS ¼Z 1

�1

Z 1

�1UT RT tlocJdndg �

Xg

xUT RT tlocJ ð8Þ

where g is the Gauss point index, x is the integration weight, U is aproperly arranged matrix of the shape functions /I (I = node index),R is the matrix of rotation from the global to the local coordinatesystems, Dtloc ¼ C locRUDu is the local traction vector increment,which is related to the increment of local displacement differencethrough a constitutive law, C loc ¼ @tloc=@Duloc , and J is the Jacobianof the transformation from the physical to the parent coordinatesof the element.

The element stiffness matrix could be obtained using

Keint ¼

@f eint

@u¼Z@Xe

UT @t@Du

UdS ¼Z 1

�1

Z 1

�1UT RT C locRUJdndg

�X

g

xUT RT C locRUJ ð9Þ

3.2. Cohesive fracturing law for foam core

The mechanical behavior of the foam is assumed to be quasi-brittle, i.e., the material is assumed to be linearly elastic until thematerial strength limit is reached, and thereafter the material is as-sumed to obey a cohesive fracturing law similar to that shown inFig. 2(a) [8]. For the size effect calculations, only the part of fractureenergy corresponding to the area under the initial tangent (labelledGfs in that figure) is important [5]. A linear cohesive formulationwith shear-tensile coupling is necessary, for the purpose of (i) cap-turing the skin wrinkling, which could be a failure mechanism inthe small size beams due to their very thin skins; and of (ii) simu-lating the curved diagonal fracture through the foam core, ob-served in the experiments.

To this end, an effective critical stress for the opening-shearfracture may be defined Eq. (10), ([11]):

reff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibr2 þ s2

pð10Þ

Here b is the ratio of fracture energies in shear and tension.Note that the cracks are properly regarded as interface cracks,

for which it is impossible to distinguish between modes I and II[31,27]. Because the skin-core ratio of tensile elastic moduli is high(about 300), one might think that the cracks could be approxi-

Fig. 2. The constitutive laws used in the calculations: (a) Softening cohesive stress-slip curve for shear cracks and its initial linear approximation (the numerical valuesfor the parameters are given in the text for various cases), (b) the assumed 1Dphenomenological elastic–plastic constitutive law at finite strains for the rubbersheets, and (c) softening law simulating local buckling (crushing) failure of polymercomposite skin under compressive stresses triggered by shear damage in thematrix.

340 F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348

mately treated as delamination cracks on a rigid substrate, forwhich modes I and II can be distinguished. However, accordingto [3], the inevitable imperfections in laminate foam sandwichesalways induce skin wrinkling within the FPZ, and that precludesmode separation even if the skin is much stiffer than the core.

Furthermore, it may be assumed that a potential for the cohe-sive stresses, Wðweff Þ (free energy density) exists, i.e.

reff ¼ brweff

wr ¼ sweff

ws ð11Þ

where wr and ws are the respective cohesive displacements in ten-sion and in shear at which the corresponding cohesive stresses be-come zero. Substitution of Eq. 11 into Eq. 10 yields the effectivecohesive displacement in terms of the cohesive displacements intension and shear:

weff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðwrÞ2=bþ ðwsÞ2

qð12Þ

Then the cohesive law may be prescribed as a relation between theeffective cohesive stress and displacements, i.e., reff ¼ reff ðweff Þ.

The evolution of both tensile stress rðwrÞ and shear stress sðwsÞcan be determined incrementally during the formation of opening-shear cohesive fracture reff ðweff Þ:DrDs ¼

rprev

sprevor

DrDs ¼

Drprev

Dsprevð13Þ

Here the subscript ‘prev’ labels the previously converged value(seeFig. 2(a), and imagine the vertical axis to be reff and the horizontalaxis to be weff ). But it is found that although this formulation con-verges for large size beams, for small size beams it might not.

This problem may be overcome by an alternative formulation,similar to that of [11], in which separate cohesive laws are pre-scribed individually for tensile and shear stresses, while the tensileand shear strength values at which crack initiates are determinedfrom the criterion

reff ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibr2 þ s2

p¼ f 0 ð14Þ

Here f 0 ¼ 1:6 MPa = initial effective stress for opening and shear.Optimal fitting of the experimental data shown in Fig. 3(a)–(c) fur-nished b ¼ 0:22 and Gfs ¼ 0:40 kN=mm. Once the crack initiates, thetensile and shear stresses are ramped down to zero, during propor-tional loading, from their values at fracture initiation, which arew ¼ wr

1 ¼ 0:28 mm and w ¼ ws1 ¼ 0:5 m. When there is pure tensile

stress, then, at the initiation of cohesive fracture, reff ¼ffiffiffibp

f 0t ¼ffiffiffibp

3:4 ¼ 1:6 MPa and, with advancing fracture, the tensile stress isramped down from 3:4 MPa to zero at w ¼ wr

1 ¼ 0:28 mm. This leadsto a purely tensile cohesive fracture with fracture energyGf ¼ f 0t wr

1=2 ¼ 0:49 N=mm, which is a value reported in [8]. Onthe other hand, when there is only shear loading, then, at crack initi-ation, reff ¼ f 0s ¼ 1:6 MPa, and, with advancing fracture, theshear stress is ramped down from 1.6 MPa to zero atw ¼ ws

1 ¼ 0:5 mm. This leads to a pure shear fracture with fractureenergy Gfs ¼ f 0s ws

1=2 ¼ 0:40 N=mm.The softening laws for tensile and shear fractures are given by

rb ¼ 1�wr

wr1r0

sb ¼ 1� jwsj

ws1

s0sgnðwsÞ ð15Þ

In the foregoing equations, wr P 0 is the normal (opening) dis-placement; wr

1 ¼ 0:28 mm is the critical normal (opening) displace-ment beyond which the cohesive stress vanishes; < x > is x if x > 0and 0 otherwise; r0 and s0 are, respectively, the tensile stress andshear stress at which the critical stress intensity factor is reached;ws is the shear displacement; ws

1 ¼ 0:5 mm is the critical shear dis-placement beyond which the cohesive shear stress vanishes; andsgnðxÞ is �1 if x < 0 and þ1 otherwise.

Assuming unloading and reloading to be linear, one may calcu-late the stresses as

r ¼minrb

prev

wrprev

wr;rb

!

s ¼minsb

prev

wsprev

ws

����������; sb�� �� !

sgnðwsÞ ð16Þ

Fig. 3. Load vs load-point displacement fits of experimental data on three differentsizes of (a) top-notched beams, (b) bottom-notched beams and (c) beams withoutnotch.

F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348 341

where rbprev or wr

prev , and sbprev or ws

prev are, respectively, the lateststress or displacement indicators of tensile damage and sheardamage.

Obviously, in this alternative formulation, a cohesive potentialdoes not exist. Nevertheless, experience shows that this formula-tion normally converges, even under highly nonproportional loads.

To simulate cracks in the foam adjacent to the skin–foam inter-face, the foregoing cohesive laws are used as the constitutive rela-

tions for the interface elements. They are also used in the interfaceelements simulating curved cracks in the foam core, which formedin Series II and III. Taking into account the combined opening-shearfracture at the skin–foam interface is found to be indispensable forfitting the test data for all the small size specimens; otherwise thecalculated response would always overestimate the data. For themedium size specimens, this is less important, and for the largesize specimens it is not important at all. This finding for cohesivecracks is analogous to the fact that the near-tip field of sharp inter-face cracks cannot be scaled and the ratio of near-tip normal andshear stresses on the crack extension line varies when the cracklength changes [31,27].

3.3. Softening laws for failure of upper composite skin

The failures of un-notched sandwich beams exhibited damagein the composite skin. The composite (IM6-G/3501-6) from whichthese beams were made was analyzed in [24,15], and the com-pressive strength of the beam was shown to depend on the shearstrength of the composite along the fiber direction, and on theinitial misalignment of the fibers. Thus, the damage in the skins,in the form of local crushing with fiber separation around theedges of the loading platen, must have been induced by a localshear failure of the composite skin, because local buckling ofthe fibers could not occur without sufficient shear damage inthe matrix.

Initially, the composite skin behaves linearly, according to atransversely isotropic stiffness matrix. It is assumed that the com-posite fails under a local shear concentration next to the edge ofthe loading platen, which gives rise to the local buckling of the fi-bers under compression. This crushing failure of the upper skin ismodelled by inserting interface elements perpendicular to thefibers.

The constitutive law for the interface elements may be writtenas (see Fig. 2(c)):

ifs P sc : rbc ¼min 1� wc

wc1

� �rc

0;rrc

� �

r ¼maxrprev

c

wcprev

wc;rbc

!ð17Þ

Here wc < 0 is the normal (crushing) displacement;wc

0 ¼ �0:025 mm is the critical normal (crushing) displacement be-yond which the cohesive stress reduces to the residual stressrr

c ¼ �20 MPa;rc0 ¼ �180 MPa = compressive stress at which the

crushing is triggered; rprevc is the latest stress indicator of crushing

damage; and sc ¼ 60 MPa = shear strength of the composite alongthe fiber direction, which takes into account the fiber misalignment[24]. The residual stress, rr

c , is generally important only when thepost-peak load-deflection curve needs to be obtained, but this isnot the case here; rr

c may be important for size effect calculationsonly if crushing in the composite becomes extensive before thepeak load is reached.

It is found that the crushing law just described is crucial forbeing able to fit the experimental data for sandwich beams withoutnotches. For small size sandwich beams, it is necessary to assume atensile failure criterion for failure of the upper skin because, at theedge of the loading platen, the tests show a sharp local curvaturecausing additional compressive and tensile stresses. As crushingof the top of the upper skin progresses, the tensile stresses at thebottom of the upper skin gradually increase and reach the tensilestrength of the composite before the peak load. At that moment,however, the tensile stress drops suddenly to zero and the beamfails. Thus the composite is assumed to fail in tension, with a sud-den drop of the normal stress to zero when the tensile strength isexhausted.

Fig. 4. (a) The force–displacement predictions of the Series III sandwich beams thatfail with only interface shear fracture; displacements are free of rubber contribu-tion, (b) predicted nominal strength of these beams and their comparison to the sizeeffect law for failure at crack initiation given in Eq. 23.

342 F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348

3.4. Simulation of notches

One possibility to simulate a notch is to mesh it as a stress-freesurface. However, because the notch faces will come into contactwith advancing deformations, a contact algorithm would also beneeded in this case. Therefore, the notches are convenientlymeshed as if they were regular interface elements. Then, the con-stitutive law for the interface elements in the notch zone is pre-scribed as

s ¼ 0;r ¼ Enotch < �wr > sgnðwrÞ ð18Þ

where s and r are, respectively, the shear and normal resistance ofthe interface element against sliding and opening. According to thisequation, no shearing resistance is provided at the notch by theinterface elements. The friction between the surfaces when theyare in contact is neglected. When the normal displacements are ten-sile, again no resistance is generated. But if the normal displace-ments are compressive, then the stress becomes very largebecause of a large penalty parameter, Enotch ¼ 104 GPa.

In the case of bottom-notched specimens, Coulomb friction ef-fects, if needed, may be taken into account by modifying appropri-ately the first of the Eq. 18. However, the friction at these notches isfound not to be significant enough to affect the load–displacementresponse.

3.5. Effect of rubber sheet under loading platen

It is well known that the compressive crushing strength of poly-mer composites along the fiber direction depends on the initial fi-ber misalignment and the shear strength of the composite alongthe fiber direction [24,15]. Thus, to prevent a local shear failureof polymer composite skins around the edges of the loading platen,rubber sheets were inserted between the loading platen and thetested beam. This measure was successful in preventing the localshear failure for all the specimens tested except for the notch-lessbeams, which exhibited shear failure at the platen edges because ofbeing able to sustain higher loads than the notched ones.

The deflection history was conveniently recorded in terms ofstroke versus applied force. It is now found that the contractionacross the rubber sheets must have significantly contributed tothe stroke measured in the medium and large size specimens,and so it must be taken into account. In the small size specimens,because of their lower failure load, the contribution of deformationin the rubber sheets to the total deformation recorded is negligible.

Common rubbers are known to exhibit no inelastic deformationand follow the Mooney–Rivlin elastic material model. However,comparison of the test results with the finite element simulationsindicates that the rubber sheets under the loading platens in themedium and large size specimens must have behaved quasi-plasti-cally before the peak load was reached. Although full explanationof this behavior is beyond the scope of this study, one may surmisethat, under vertical compression, these sheets exhibited, due totheir negligible bulk compressibility, large lateral expansions,which caused them to slip in contact with both the platens andthe laminate skins. The lateral expansion due to slipping may allowthe rubber sheet thickness to contract as if it were plastic (i.e.,although the slip of rubber on the microscale consists of propaga-tion of tiny ripples of separation, the resulting effect is roughlysimilar to a plastic deformation).

The apparent quasi-plastic behavior of the rubber sheets is rep-resented, in the phenomenological sense, by a simple one-dimen-sional elastic plastic constitutive law, as shown in Fig. 2(b). Toidentify this law from the test data, the dimensions of the loadingplaten and the thickness of the sheets were considered. A linearelastic law with Erubber ¼ 8:4 MPa was found to suffice to fit theexperimental data in the elastic range. The engineering stress at

yield that, allowing the best fit of the experimental data of allthe series (a total of 9 test data), is Ty ¼ �0:46 MPa. Thus, the com-pressive constitutive law of the rubber sheets may be expressed as

T ¼ ErubberDu=t0 if 0 P Du=t0 P �0:055T ¼ A lnð�Du=t0Þ þ Bð1þ Du=t0Þ þ C if Du=t0 < �0:055 ð19Þ

where A ¼ �0:4943 MPa; B ¼ �0:6549 MPa; C ¼ �1:3 MPa; t0 =initial thickness of the rubber sheet, Du = its compressive displace-ment, and T = engineering stress in compression.

3.6. Size effect law for unnotched sandwich beams failing by interfaceshear fracture

Although skin failure is one important phenomenon for sand-wich structures, this study is focussed only on the cohesive shearfailure at the skin–foam interface and its implications for size ef-fect on the nominal strength of the sandwich beams. Now thatwe have calibrated the finite element model, we try to predictthe failure loads of Series III specimens when they fail by interfaceshear fracture only. The finite element results for such failures areshown in Fig. 4(a) and (b).

Fig. 5. (a) Oblique view of deformed mesh and shear stress distribution at theinterfaces of the large size top-notched beam, (b) oblique view of deformed meshand shear stress distribution at the interfaces of the medium size top-notchedbeam, (c) oblique view of deformed mesh and shear stress distribution at theinterfaces of the small size top-notched beam.

F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348 343

A comparison of the predicted failure loads shown in Fig. 4(a)with the measured ones shown in Fig. 3(c) indicates that, whenonly the interface shear failure is considered, the failure loads aremuch larger than the measured ones, especially in the small sizespecimens. Thus, it is necessary to recalibrate the size effect lawproposed in [6].

As observed in finite element calculations, the notch-less beamsfail as soon as the fracture process zone (FPZ) forms at the skin–foam interface. Approximating quasibrittle fracture according tothe equivalent linear elastic fracture mechanics (LEFM), one canexpress the nominal strength as [7,2,5]

rN ¼

ffiffiffiffiffiffiffiffiffiffiffiffiEGfs

gðaÞh

s; a ¼ a

h; a ¼ a0 þ cf ð20Þ

where E is Young’s modulus; a0 is the length of the stress-free crackat maximum load; a is the length of the equivalent crack at maxi-mum load; a is the dimensionless crack length; and cf is the halflength of FPZ, which is assumed to be size independent. In the fore-going equation, gðaÞ ¼ k2ðaÞ where kðaÞ ¼ KIb

ffiffiffihp

=P in which KI isthe stress intensity factor; P is the failure load; and b, h are the beamwidth and depth. Normally gðaÞ is calculated numerically for spec-imens under complex boundary conditions, and analytically forsimpler ones. This study uses an approximation of gðaÞ obtainedin [6] analytically, based on enriched sandwich beam theory:

gðaÞ ¼ 175;200a3 � 536;200a2 þ 686;600a ð21Þ

Because failure happens as soon as the FPZ develops its full size, thefirst three terms of the Taylor series expansion of gðaÞ are needed(the first two terms do not suffice because the first term gða0Þ ¼ 0at a0 ¼ 0);

gðaÞ � g0ða0Þcf

hþ 1

2g00ða0Þ

cf

h

� 2ð22Þ

which is substituted into Eq. 20. Next, the resulting expression ismodified so that it provides the simplest asymptotic matching ofthe small-size and large-size nominal stress expressions up to thefirst two nonzero terms; this yields [5]:

rN ¼ r1N 1þ hb

h

� �ð23Þ

in which the constants r1N ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiEGfs=ðg0ð0Þcf Þ

pand

hb ¼ �cf g00ð0Þ=ð4g0ð0ÞÞ.The optimal fit of the predicted nominal stress values is shown

in Fig. 4(b). The optimal values of fracture energy and the halflength of FPZ are found to be Gfs ¼ 987:4 kN=mm andcf ¼ 4:85 mm. These values are, as expected, much larger thanthose obtained from the experimental failure loads of these beams,which were reported in [6] to be Gfs ¼ 240:4 kN=mm andcf ¼ 2:9 mm.

4. Results and discussion

The finite element calculations were run up to the peak loadand a little beyond, using the displacement control, under condi-tions identical to the experiments. In each loading step, after con-vergence, the principal stresses were calculated at all the Gausspoints to check that the maximum tensile, maximum compressiveand maximum shear stresses are not smaller than the values sup-plied by the manufacturer, which were 3:4� 1:0 MPa; 2:0�0:35 MPa and 1:6� 0:2 MPa, respectively [18].

The dimensions of the test specimens of three different sizes ofseries I, which are taken from [6], are shown in Fig. 1(a). The beamsin this series have notches at the top interface between the skinand the foam. The test data reported in that study and the corre-sponding finite element simulations obtained in this study are

shown in Fig. 3(a). The fits are excellent, for specimens of all thesizes. In Fig. 5(a)–(c), the boundary conditions, the deformed con-figuration and the stress state at the foam–skin interface at peakload are shown for all the sizes of beams with notches at the topinterface. In the simulation of series I tests, the stresses in the rub-ber sheet of the largest, medium and smallest specimens are foundto be about 1.21 MPa, 1.03 MPa and 0.5 MPa, respectively.

The dimensions of the test specimens of three different sizes ofseries II, taken from [6], are shown in Fig. 1(b). The beams in thisseries had notches at the bottom interface between the skin andthe foam. The test data reported in that study and the correspond-ing finite element simulations obtained in this study are shown in

344 F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348

Fig. 3(b). For all the sizes, the fits are excellent. In Fig. 6(a)–(c), theboundary conditions, the deformed configuration and the state ofstress at both the foam–skin interface and the diagonal curvedcracks at peak load are shown for all the sizes of beams withnotches at bottom interface. In the simulation of series II tests,the stresses in the rubber sheet of the largest, medium and small-est specimens at maximum load are again found to be about1.0 MPa, 1.1 MPa and 0.54 MPa, respectively.

The dimensions of the test specimens of three different sizes ofseries III, taken from [6], are shown in Fig. 1(c). The beams in thisseries had no notches. The test data reported in that study and thecorresponding finite element simulations obtained in this studyare shown in Fig. 3(c). The fits are again excellent for specimens

Fig. 6. (a) Oblique view of deformed mesh and shear stress distribution at theinterfaces of the large size bottom-notched beam, (b) oblique view of deformedmesh and shear stress distribution at the interfaces of the medium size bottom-notched beam, (c) oblique view of deformed mesh and shear stress distribution atthe interfaces of the small size bottom-notched beam.

of all the sizes. In the simulations, the crushing of the skins hadto be taken into account in accordance with the constitutive lawshown in Fig. 2(c). In addition, for the small-size specimen, tensilefailure of the skins had to be considered in the form of a suddenstress drop upon reaching the tensile strength. In Fig. 7(a)–(c),the boundary conditions, the deformed configuration and the stateof stress at both the foam–skin interface and the diagonal curvedcracks at peak load are shown for all the specimen sizes of thebeams without notch. In the simulation of series III tests, the stres-ses in the rubber sheet of the largest, medium and smallest speci-mens at maximum load were found to be about 1.3 MPa, 1.07 MPaand 0.6 MPa, respectively.

To achieve the fits shown in Fig. 3(a)–(c), it is essential to con-sider a combined opening-shear cohesive fracture. When the com-

Fig. 7. (a) Oblique view of deformed mesh and shear stress distribution at theinterfaces of the large size beam without notch, (b) oblique view of deformed meshand shear stress distribution at the interfaces of the medium size beam withoutnotch, (c) oblique view of deformed mesh and shear stress distribution at theinterfaces of the small size beam without notch.

F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348 345

bined opening-shear nature of the fracture at the skin–foam inter-face is neglected, the calculated load–displacement response al-ways overestimates the experimental data in small and mediumsize specimens of all the three series. The numerical simulationof the large-size specimens, on the other hand, is found capableof fitting the experimental data equally well, regardless of whethera cohesive opening-shear fracture, or a purely cohesive shear frac-ture, is employed. Consequently, to capture the combined opening-shear effects, it is important to use a fine mesh for small-size spec-imens. For large-size specimens, on the other hand, relativelycoarse meshes can be employed. The meshing requirements forthe medium-size specimens are in between.

According to the Figs. 5 and 7 the notched specimens developrelatively longer and longer cracks along the notched interface asthe specimen size changes from large to small. Diagonal curvedcracks were allowed to form in the foam core for Series II and Ser-ies III specimens, to find out whether these cracks could developbefore or after the peak load. In the simulations of Series II speci-mens, these cracks were found to form only in the post-peak re-gime when the axial stresses get transferred from the skins tothe foam core, due to a large reduction of interface shear resis-tance. By contrast, simulation of the notch-less specimens in SeriesIII showed limited fracture propagation at the skin–foam interface.

According to the simulations of the large and medium size spec-imens, the interface fracture initiates at the edges of loading plat-ens near the support, at which skin crushing takes place. For asmaller specimen size, the initiating crack spreads over a largerand larger portion of the interface, which is explained by anincreasing ratio of the FPZ size to specimen size.

Note the abrupt curvature change at the crushed skins, as sim-ulated by the interface elements inserted perpendicular to the fiberdirection in the upper skins. The stress state in the diagonal curvedcracks at peak load indicates that the diagonal fracture happensonly after the peak load.

It is found that failure in the small-size specimens of Series I andII developed before the peak load, and only within the foam coreand the interface. In all the larger specimens of Series I and II,the beams are found to have failed by interface shear fracture only,which is manifested by quasibrittle behavior. However, the addi-tional failure mechanisms vary among the small-size specimensfrom one series to the other. Fig. 8(a) and (b) depict the minimum

Fig. 8. (a) The pore collapse mechanism of failure of the foam core in the small size tobottom-notched beam.

and maximum principal stress distributions, respectively. Underuniaxial compression, the pore collapse in foam is known to beenergetically more favorable than the shear failure [8]. Thus, inthe small-size specimens of top-notched beams, pore collapse inthe foam core took place, after a large interface fracture, near theedges of the loading platen, as shown in Fig. 8(a). In the bottom-notched small-size specimens, again after the growth of very longinterface fracture, distributed tensile cracks are found to haveformed in the bottom half of the foam core, as shown in Fig. 8(b).

When the additional failure mechanisms are blocked in the sim-ulations, the maximum increase in the peak load of the top-notched and bottom-notched small size specimens is found to beless than 6% and 11% respectively. Thus, the effect of this changeon the size effect law proposed for top-notched and bottom-notched specimens in [6] must be small.

In the simulations of notch-less beams, it is found that the addi-tional failure mechanism that must have intervened in all thebeams is the failure of skin. Modeling of failure of the compositeskin in the form of compression crushing of fibers is sufficient toachieve the peak load in the large- and medium-size specimensbut, in the small-size specimen, tensile failure of the skin mustbe considered. Except at the two interfaces, the foam core remainselastic everywhere in all three specimens until the peak load isreached. When this additional failure mechanism is ignored, theincrease in computed maximum load is relatively very large forthe small-size specimens and small for the medium- and large-sizespecimens. Consequently, the size effect law proposed in [6] forfailure at crack initiation needs to be revised.

Fig. 4(a) and (b) depict the predicted failure loads of Series IIIbeams under interface shear fracture only, and also the size effectfit to the predicted nominal stress values. In the fit, the same func-tion gðaÞ as determined by the enriched sandwich beam analysis[6] is used. As expected, the fracture energy and half-length ofFPZ increase from Gfs ¼ 240:4 kN=mm and cf ¼ 2:9 mm toGfs ¼ 987:4 kN=mm and cf ¼ 4:85 mm.

One important question to answer using the present computa-tional framework is whether the skin wrinkling is an additionalfailure mechanism in the small-size beams, in which the skin isvery thin. To this end, it is necessary to consider an opening-shearcohesive fracture law and geometrical nonlinearity. It transpiresthat no skin wrinkling could have taken place in the small-size

p-notched beam, (b) the distributed tension failure of foam core in the small size

346 F.C. Caner, Z.P. Bazant / Composites: Part B 40 (2009) 337–348

beams with notches, for three reasons: (i) The tensile strength offoam at the skin–foam interface is too high for crack initiation;(ii) once the cohesive shear fracture begins, in accordance withthe combined opening-shear fracture criterion, the tensile strengthat the damaged interface drops, but it drops first at the edgeswhere the compressive stress is the smallest; and (iii) by the timethe shear fracture approaches the midspan, the length of the skinsubjected to high compressive stress becomes too small. Neithercan skin wrinkling happen in the small-size specimens without anotch, because (i) skin crushing causes premature failure, and (ii)the bond between the foam and the skin remains too strong forany wrinkling to take place during crack initiation failure beforethe peak load is reached.

5. Conclusions

1. Failure of laminate-foam sandwich beams of various sizescan be realistically simulated by geometrically nonlinearfinite element analysis in which cohesive cracks are repre-sented by zero-thickness interface elements, provided thata combined cohesive opening-shear softening and skincrushing are taken into account.

2. It is important to take into account the deformation of therubber sheets placed under the loading platens to preventlocal shear failure of the laminate skin. Although rubber isperfectly elastic, the one-dimensional behavior of these rub-ber sheets, accounting for the effect of lateral expansion, hadto be represented as elastic–plastic, in order to fit the exper-imental data for the medium and large size specimens. Thecause of this seemingly plastic behavior under the loadingplatens might be the volume incompressibility of the rubbersheets and their low shear modulus, along with ripplingslippage at their contact surfaces. For the smallest sizebeams, the effect of the deformation of rubber sheets isnegligible.

3. The calculations confirm that failure of sandwich plates con-sisting of fiber composite skins and PVC foam core exhibits asignificant size effect. This is true not only for the notched(or damaged) sandwich plates but also for the notch-less ones.

4. Consequently, it is unrealistic to predict the load capacity oflarge sandwich structures on the basis of material failure cri-teria expressed only in terms of stresses and strains, whichare inherently incapable of capturing non-statistical sizeeffects. The concepts of material strength and yield limitdo not apply. Instead, the load capacity of large sandwichstructures must be determined on the basis of the cohesivecrack model or nonlocal continuum damage models.

5. Based on optimum fitting of failure test data, the shear frac-ture energy of the PVC foam is estimated to be 0.40 N/mm,which is about 82% of the opening mode fracture energy [8].

6. Aside from interface shear fracture on approach to the peakload, further failure mechanisms operate in small-size beams,but not large ones. They include compressive crushing of thefoam near the edges of the loading zone in top-notchedsmall-size beams, and distributed tensile fracture of the foamin the bottom-notched small size beams. However, thesemechanisms do not have a strong effect because, when theyare excluded from simulations, the increase in the computedfailure load of small size beams is less than 6%.

7. The failure load of notch-less specimens is drasticallyaffected by crushing of the skin, which can be modelled byinterface elements inserted perpendicularly to fibers nearthe edges of the loading platen, using a compressivestress-compressive displacement law with linear softeningterminating with a horizontal residual stress and triggeredby the shear damage in the matrix.

8. The experimental maximum loads of notch-less beams canbe closely predicted considering only shear fracture in thefoam adjacent to the interface. These data show that thesize effect at crack initiation is strong. By fitting the sizeeffect law to these data, one gets, for shear fracture inthe foam near the interface with skin, the FPZ half-lengthcf ¼ 4:85 mm, and the fracture energy Gfs ¼ 987 kN=mm.The latter value is 2� 106-times larger than the Mode Ifracture energy of the foam material alone [8]. This maybe explained by the composite action of foam and skinin which the external loads are resisted by axially verystiff skins.

9. Numerical simulations show that the curved diagonal frac-ture observed to run across the foam core cannot affect thepeak load and thus must have formed in the post-peakregime of the tested beams.

10. Geometrically nonlinear finite element simulations withopening-shear cohesive interface fracture calibrated to fitexperimental data reveal that the skin wrinkling, whichmight be suspected to occur in small-size beams due to theirvery thin skins, has not actually taken place.

Acknowledgements

Financial support from the Office of Naval Research undergrants N00014-91-J-1109 and N00014-07-I-0313 to NorthwesternUniversity is gratefully acknowledged. Partial support from GrantBIA2006-12717 from MEC-Madrid is also acknowledged.

Appendix A. Summary of previous size effect tests of sandwichbeams with foam core

As reported in detail in [6], the cores were made of closed cellpolyvinylchloride (PVC) foam with mass density 100 kg=m3

(25.4 mm thick sheets of Divinycell H100, procured from Diab-group, Inc.). The properties of the foam (as specified by the sup-plier) were as follows [18]: tensile elastic modulus 130� 25 MPa,tensile strength 3:4� 0:4 MPa, compressive elastic modulus135� 20 MPa, compressive strength 2:0� 0:35 MPa, elastic shearmodulus 35� 7 MPa, and shear strength 1:6� 0:2 MPa. The sand-wich beams fabricated had the same width, b ¼ 25:4 mm, for allthe three sizes (in order to eliminate any possible width effect).All of the cores were cut from the same sheet of foam [6]. The testsand finite element analyses indicated that there was a variation inthe foam properties, though within the limits specified by themanufacturer. The thickness t of the skins was scaled in proportionthe core depth, c.

For series I and II, notches were cut in the foam, symmet-rically at both ends of beam, as close as possible to the inter-face with the top or bottom skin, but without cutting into theskin, and without baring it (Fig. 1(a) and (b)). The distance aof each notch tip from the support axis was equal to coredepth c. Each notch tip was sharpened by a razor blade of0.25 mm thickness.

The purpose of making the notches was to clarify the effect oflarge pre-existing cracks or damage zones, and to force the fractureto develop at a certain pre-determined location, homologous (geo-metrically similar) for all the sizes. The notches ensured the ab-sence of statistical Weibull-type contribution to the size effect.The choice of beam and notch proportions was also guided bythe need to prevent significant core indentation, which wouldcomplicate interpretation of the results. Furthermore, the notchesmade it possible to obtain shear fracture for a relatively smalllength-depth ratio of the test beams. All the beams were subjectedto three-point loading.

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A.1. Test series I, with fiberglass-epoxy skins and notches at topinterface

According to [6], the skins of beams of series I consisted of acured woven glass-epoxy composite (FS-12A, procured from Aero-space Composite Products, as 0.38 mm thick sheets). The followingproperties were obtained through material tests: longitudinal elas-tic modulus 30 GPa, transverse modulus 28 GPa, transversestrength 300 MPa, in-plane shear modulus 5 MPa, and in-planeshear strength 90 MPa. The depths of the foam cores werec ¼ 10:2; 40:8 and 163.2 mm (Fig. 1(a)–(c)). The scaling of the skinthicknesses was achieved by bonding 25.4 mm wide compositestrips—1 strip for the smallest beams, 4 strips for the mediumbeams, and 16 strips for the largest beams. The L=h ratio was 6:1for all the beams (and thus the effective lengths of the beams wereL ¼ 61:2; 244:8 and 979.2 mm). All the beams of series I wereloaded in an Instron 5000 universal testing machine.

The local failure under the corners of the loading platens is asensitive aspect. It could be so extensive as to control the maxi-mum load. To avoid this problem, two separate rubber sheets wereplaced on top of the skin. Furthermore, between these sheets andthe metallic platen (aluminum for series I, and steel for series IIand III), two laminate plates (consisting of the same material asthe skins) were inserted (having dimensions 10� 25:4�6 mm; 40� 25:4� 6 mm and 160� 25:4� 6 mm for series I, and17� 25:4� 4:8 mm; 51� 25� 4:8 mm and 154� 25:4� 4:8 mmfor series II and III). Observations during the series I tests showedthat a small crack grew in the top laminate plate from the edgeof the platen, but did not extend into the second laminate plate.

Fig. 3(a) shows, for all three sizes, the typical records of theload-deflection curves up to failure. They show that the largerthe specimen, the more brittle its response. For example, in thelargest specimen, crack propagation becomes unstable and theload suddenly drops right after the peak load, while in the smallestspecimen, one can see a period of gradual post-peak softening. Inthe largest specimen, the loss of crack growth stability after thepeak was manifested by a loud noise, while in the medium sizespecimen only quiet sound emissions during the gradual postpeakdecline of load were heard.

A.2. Test series II, with carbon-epoxy skins and notches at bottominterface

The foam cores in series II specimens were the same as in Series I[6]. However (for no other reason than convenient availability), a dif-ferent material was used for the skins—unidirectional carbon -epoxy(IM6-G/3501-6, provided by Hexcel, Inc., in a 12 inch wide roll,Young’s moduli E1 ¼ 172 GPa and E2 ¼ 9:5 GPa, shear modulusG12 ¼ 5:7 GPa and Poisson ratios m12 ¼ 0:28 and m21 ¼ 0:02). The pre-preg material was cut into 12� 12 inch sheets for the small andmedium size specimens, and 34� 10 in. sheets for the large speci-mens. Then, for small, medium and large specimens (Fig. 1(b)),respectively, 3, 9 and 27 layers of prepreg sheets were laid up. The12� 12 inch laminates were cured in a compression-laminationpress (Tetrahedron MTP-14), while the 34� 10 inch laminates werecured in an autoclave (McGrill). The curing process suggested by themanufacturer for the 3501-6 epoxy matrix was followed (it involveda maximum curing temperature of 350�F and a maximum curingpressure of 75 psi). After curing, the laminates were cut into3:67� 1:0;11:0� 1:0 and 33:0� 1:0 inch strips, by means of a dia-mond saw. The thicknesses of the ½0�3, ½0�9 and ½0�27 laminate skinswere 0.021, 0.063 and 0.189 inch (0.53, 1.60 and 4.80 mm) for thesmall, medium and large specimens, respectively. The foam (Diviny-cell H100) was cut into 3:67� 0:33� 1:0;11:0� 1:0� 1:0 and33:0� 3:0� 1:0 inch ð93:2� 8:4� 25:4;279:6� 25:4� 25:4;839:0� 76:2� 25:4 mmÞ blocks for the small, medium and large

specimens, respectively. The L=c ratio of these beams was 9:1,and thus the effective beam lengths were 3, 9 and 27 inch, or76.2, 228.6 and 685.8 mm, respectively (Fig. 1(b)).

The skins and foam cores were bonded together by epoxy adhe-sive and were cured in a vacuum bag for 8 hours.

The small and medium specimens were tested under three-point bending in a universal testing machine (MTS, 20 kip), whilethe large specimens were tested in a larger machine (MTS 220kip) because of their greater dimensions rather than strength.The loading rate was chosen so as to reach the peak load of thespecimens of all the sizes within about 3 minutes [6].

For all the specimens, without exception, the final failure modeobserved after the load dropped included diagonal tensile fracturewhich propagated through the foam [6]. Whether the diagonal ten-sile fracture in the foam occurred after, at, or just before the peak loadwas not recorded. The post-peak behavior confirmed that the largerthe specimen, the greater the brittleness of response. The load-deflection curves for all three specimen sizes are shown in Fig. 3(b).

A.3. Test series III, with carbon-epoxy skins and no notches

As reported in [6], the beams of test series III were the same asin series II, except that the foam block dimensions were 3:67�0:25� 1:0; 11:0� 0:75� 1:0; 33:0� 2:25� 1:0 in. or 93:2� 6:4�25:4; 279:6� 19:0� 25:4; 839:0� 57:2� 25:4 mm, which meansthat the ratio of L=c was 12:1. In addition, the notches were absent(Fig. 1(c)). The scaling ratios of the length and depth dimensions ofall the beams were 1:3:9 (Fig. 1(c)).

In series III, the failure modes varied. In the small beams, nodiagonal shear fracture was observed and the failure was causedby compressive fracture of the upper skin. This mode of failurewas also observed in some medium and large beams, but in mostof these beams the interface crack branched into a diagonal tensilecrack which crossed the foam core [6]. The measured load-deflec-tion curves for the beams of all three sizes are shown by the pointsin Fig. 3(c).

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